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%% lecture17.tex
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%% Started on  Thu Jan  5 08:21:15 2012 alex
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\exercises
Given two disjoint connected $n$-manifolds $M$ and $N$, a
connected $n$-manifold $M\connectSum{N}$, their connected
sum\index{Connected Sum}\index{Sum!Connected}, can be constructed
by deleting the interiors of small closed balls $B_1\propersubset
M$ and $B_2\propersubset N$ and identifying the resulting
boundary spheres $\partial B_1$ and $\partial B_2$ via some 
homeomorphism between them.
\begin{xca}
Assuming that $M$ and $N$ are closed orientable manifolds prove
that $H_k(M\connectSum{N}; \ZZ)$ is isomorphic to direct sum of
$H_k(M; \ZZ)$ and $H_k(N; \ZZ)$ for $0 < k < n$.
\end{xca}
\begin{xca}
Let $M$ denote a closed $n$-dimensional connected orientable
manifold. Assuming that we know the cohomology of $M$ calculate
the cohomology with compact supports of $M \setminus A$ where
\begin{enumerate}
\item $A$ is a finite subset of $M$,
\item $A$ is a union of boundary spheres $\bdry B_1$,\dots,
  $\bdry B_n$ of non-overlapping small closed balls $B_1$, \dots,
  $B_n$ in $M$.
\end{enumerate}
\end{xca}
